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Exact colinearity of centroids of iterated midpoint hexagons

Jack Edward Tisdell

Abstract

We study the iteration that replaces a planar hexagon by the hexagon formed by joining the midpoints of consecutive edges. While this iteration quickly drives any polygon toward a point and their shapes asymptotically regularize, we show a stronger and unexpected rigidity holds for hexagons: from the second iterate onward, the centroids of the filled hexagons all lie exactly on a fixed line. This exact colinearity reflects a special algebraic feature of the hexagonal case and does not hold generally for any other polygons.

Exact colinearity of centroids of iterated midpoint hexagons

Abstract

We study the iteration that replaces a planar hexagon by the hexagon formed by joining the midpoints of consecutive edges. While this iteration quickly drives any polygon toward a point and their shapes asymptotically regularize, we show a stronger and unexpected rigidity holds for hexagons: from the second iterate onward, the centroids of the filled hexagons all lie exactly on a fixed line. This exact colinearity reflects a special algebraic feature of the hexagonal case and does not hold generally for any other polygons.

Paper Structure

This paper contains 4 sections, 4 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The midpoint iteration and its eigenstructure in the space of hexagons
  3. The centroids
  4. The colinearity phenomenon is unique to hexagons

Key Result

Theorem 1

From the second iterate onward (that is, excluding $G_0$), the points $G_n$ ($n\ge 1$) lie on a fixed line and converge eventually monotonically along it to the centroid of the vertices of $P_0$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Starting with any planar hexagon and iteratively forming the hexagon by joining the midpoints of consecutive sides, all the centroids except possibly the first lie on a common line.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Proposition 4
  • proof